The locus of points at a given distance from a given point is a circle. What can we say about Class 9

The locus of points at a given distance from a given point is a circle. What can we say about Class 9

Question 1.

The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points? (Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.) Class 9

Answer:

The locus of points equidistant from two given points is the perpendicular bisector of segment joining those points.

Proof:

Let M be the midpoint of AB and P be any point on the perpendicular bisector of AB.

In ∆PAM and ∆PBM,

AM = BM ..[M is the midpoint of AB]

PM = PM ....[Common side]

∠PMA = ∠PMB = 90° ...[PM ⊥ AB]

By SAS congruence, ∆PAM ≅ ∆PBM

∴ PA = PB

Thus, any point on the perpendicular bisector is equidistant from A and B.

Question 2.

How many circles pass through two points on a plane? Class 9

Answer:

Infinitely many circles pass through two given points on a plane.